hi Wayne,
There are many different Aikido organizations, and with all the differences the name of the techniques vary. Udekime nage and udegarami sometimes interchange in Aikido dojo, also some dojos don't have udekime or udegarami. Kata-gatame on the otherhand I don't think exist in most dojos, it is a judo technique and so thus udegarami generally. But, I will take a crack at the name nonetheless:

Udekime - figure-four armlock, that looks like a Shiho-nage except that nage and uke's inside arm touches and your holding your own wrist while ukes elbow is bent. If it is a right punch you go to the outside and hold his right by your right hand from the outside, after securing right wrist by right hand you grab your own wrist by your left then bend the elbow to ukes back.

Udegarami - Kaiten-nage position except that you are locking his arm in figure-four position. If mune-tsuki apply kaiten by having your left hand hold his right arm then reaching your left wrist by your right from the back of uke while elbow is bended, the uke can then go straight to the ground while you are locking his elbow and shoulder.
This technique works better in grappling position when your opponent is pinned to the ground while you are applying the lock to the 90 degree bent of his arm.

Kata gatame - is pinning the persons arm close to his face while your smothering him by your arm in-front of him like doing the triangle choke by the arms, your heads are almost back to back. This is actually a pin and submission hold. I have seen this perform as a throw by ninjutsu guys and in grappling as a pin or choke but never in Aikido.

I hope that I made some things clear or maybe I muddled things up, anyway thats the way I see it.
Let's hear from other guys.